function [cost,grad] = sparseAutoencoderCost(theta, visibleSize, hiddenSize, ...
                                             lambda, sparsityParam, beta, data)

% visibleSize: the number of input units (probably 64) 
% hiddenSize: the number of hidden units (probably 25) 
% lambda: weight decay parameter
% sparsityParam: The desired average activation for the hidden units (denoted in the lecture
%                           notes by the greek alphabet rho, which looks like a lower-case "p").
% beta: weight of sparsity penalty term
% data: Our 64x10000 matrix containing the training data.  So, data(:,i) is the i-th training example. 
  
% The input theta is a vector (because minFunc expects the parameters to be a vector). 
% We first convert theta to the (W1, W2, b1, b2) matrix/vector format, so that this 
% follows the notation convention of the lecture notes. 

W1 = reshape(theta(1:hiddenSize*visibleSize), hiddenSize, visibleSize);
W2 = reshape(theta(hiddenSize*visibleSize+1:2*hiddenSize*visibleSize), visibleSize, hiddenSize);
b1 = theta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize);
b2 = theta(2*hiddenSize*visibleSize+hiddenSize+1:end);

% Cost and gradient variables (your code needs to compute these values). 
% Here, we initialize them to zeros. 
cost = 0; %#ok<NASGU>
W1grad = zeros(size(W1)); 
W2grad = zeros(size(W2));
b1grad = zeros(size(b1)); 
b2grad = zeros(size(b2));

%% ---------- YOUR CODE HERE --------------------------------------
%  Instructions: Compute the cost/optimization objective J_sparse(W,b) for the Sparse Autoencoder,
%                and the corresponding gradients W1grad, W2grad, b1grad, b2grad.
%
% W1grad, W2grad, b1grad and b2grad should be computed using backpropagation.
% Note that W1grad has the same dimensions as W1, b1grad has the same dimensions
% as b1, etc.  Your code should set W1grad to be the partial derivative of J_sparse(W,b) with
% respect to W1.  I.e., W1grad(i,j) should be the partial derivative of J_sparse(W,b) 
% with respect to the input parameter W1(i,j).  Thus, W1grad should be equal to the term 
% [(1/m) \Delta W^{(1)} + \lambda W^{(1)}] in the last block of pseudo-code in Section 2.2 
% of the lecture notes (and similarly for W2grad, b1grad, b2grad).
% 
% Stated differently, if we were using batch gradient descent to optimize the parameters,
% the gradient descent update to W1 would be W1 := W1 - alpha * W1grad, and similarly for W2, b1, b2. 
% 

m = size(data,2);
rho_hat = zeros(size(b1));

% calculate sparsity derivative
for i=1:m,
    % feedforward
    a1 = data(:,i);
    z2 = W1*a1 + b1;
    a2 = sigmoid(z2);
    
    rho_hat = rho_hat + a2;
end
rho_hat = rho_hat/m;
sparsity_deriv = beta*...
    (-sparsityParam./rho_hat + (1-sparsityParam)./(1-rho_hat));

% calculate deltas and gradients
cost_err = 0;
for i=1:m,
    % feedforward
    a1 = data(:,i);
    z2 = W1*a1 + b1;
    a2 = sigmoid(z2);
    z3 = W2*a2 + b2;
    a3 = sigmoid(z3);
    
    % cost due to error
    err = a3 - a1;
    cost_err = cost_err + err'*err/2;
    
    % deltas
    delta3 = err.*a3.*(1 - a3);
    delta2 = (W2'*delta3 + sparsity_deriv).*a2.*(1 - a2);
    
    % gradients
    W2grad = W2grad + delta3*a2';
    W1grad = W1grad + delta2*a1';
    b2grad = b2grad + delta3;
    b1grad = b1grad + delta2;
end
W2grad = W2grad/m + lambda*W2;
W1grad = W1grad/m + lambda*W1;
b2grad = b2grad/m;
b1grad = b1grad/m;

KLdiv = sparsityParam*log(sparsityParam./rho_hat) + ...
    (1 - sparsityParam)*log((1 - sparsityParam)./(1 - rho_hat));

cost_err = cost_err/m;
cost_weights = lambda/2*(sum(W1(:).^2) + sum(W2(:).^2)); % w regularization
cost_sparse = beta*sum(KLdiv); % induce "sparsity"

cost = cost_err + cost_weights + cost_sparse;

%-------------------------------------------------------------------
% After computing the cost and gradient, we will convert the gradients back
% to a vector format (suitable for minFunc).  Specifically, we will unroll
% your gradient matrices into a vector.

grad = [W1grad(:) ; W2grad(:) ; b1grad(:) ; b2grad(:)];

end

%-------------------------------------------------------------------
% Here's an implementation of the sigmoid function, which you may find useful
% in your computation of the costs and the gradients.  This inputs a (row or
% column) vector (say (z1, z2, z3)) and returns (f(z1), f(z2), f(z3)). 

function sigm = sigmoid(x)
  
    sigm = 1 ./ (1 + exp(-x));
end

